Question

Repeated square roots Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}}\), for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\) c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}},}\) where \(p>0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)

Short answer

Answer: The exact limit of the given expression is the golden mean, which is \(\frac{1+\sqrt{5}}{2}\).

Step-by-step solution

Define the recurrence relation and interpret the expression as a limit

Our given expression is \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}\). We can define a sequence of numbers \({a_n}\) using the following recurrence relation: $$a_{0}=1, a_{n+1}=\sqrt{1+a_{n}}, \quad \text{for} \quad n=0,1,2,3, \ldots$$ The expression is the result of applying the recurrence relation repeatedly. The value of the expression can be interpreted as the limit of the sequence \({a_n}\) as \(n\rightarrow\infty\), provided the limit exists: $$\lim_{n\rightarrow\infty} a_{n}$$

Calculate the first five terms of the sequence

Using our recurrence relation, let's calculate the first five terms of the sequence: \(a_0 = 1\) \(a_1 = \sqrt{1+a_0} = \sqrt{1+1} = \sqrt{2}\) \(a_2 = \sqrt{1+a_1} = \sqrt{1+\sqrt{2}}\) \(a_3 = \sqrt{1+a_2} = \sqrt{1+\sqrt{1+\sqrt{2}}}\) \(a_4 = \sqrt{1+a_3} = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{2}}}}\) So the first five terms of the sequence are: \(1, \sqrt{2}, \sqrt{1+\sqrt{2}}, \sqrt{1+\sqrt{1+\sqrt{2}}}, \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{2}}}}\)

Estimate the limit of the sequence and compare it with the golden mean

By calculating the first few terms numerically, we can observe that the sequence seems to converge to a value close to the golden mean, which is \((1+\sqrt{5})/2 \approx 1.61803398875\). This is only an estimate and not an exact value.

Determine the exact limit, assuming it exists

Assuming the limit of the sequence exists, we can set: $$\lim_{n\rightarrow\infty}a_n = L$$ Then, we can use the method of Example 5 (not provided), which states: $$L = \sqrt{1+L}$$ To solve for L: $$L^2 = 1+L$$ $$L^2-L-1=0$$ Solving this quadratic equation, we obtain two values: $$L=\frac{1\pm\sqrt{5}}{2}$$ Since we are looking for a positive value, we discard the negative solution and keep the positive one, which is the golden mean: $$L=\frac{1+\sqrt{5}}{2}$$ This is the exact value of the limit of the sequence.

Analyze the expression for different values of p and determine if it has a limit for all positive values of p

For the expression $$\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}},$$ where \(p>0\), we can analyze the limit of the sequence for various positive values of p. To determine if the limit exists for all positive values of p, we would need to establish an expression for the limit in terms of p and check its convergence properties. If it converges for all positive values of p, then the limit exists for all positive values of p. In practice, we can test out various values of p and see if the expression seems to approach a limit. To be conclusive, however, we would need a more rigorous mathematical argument or proof.

Key concepts

Recurrence Relation

In studying the repeated square roots problem, we encounter a special type of sequence known as a recurrence relation. This concept involves defining each term in a sequence with respect to the one before it. Specifically, for our sequence \( {a_n} \), we start with an initial term \( a_0 = 1 \) and then each term after that is formed by the relationship \( a_{n+1} = \sqrt{1+a_n} \). This serves as a mathematical foundation for building sequences in an iterative fashion, allowing us to approach complex expressions through a series of simpler steps.

A key aspect of recurrence relations is understanding how they unfold, a concept that's integral to computer algorithms, financial models, and various fields of mathematics. When presented visually, such sequences can create intricate patterns and, in our case, help visualize the convergence towards a limit.

Convergent Sequence

The repeated square roots exercise also introduces us to the idea of a convergent sequence. A sequence is said to converge if its terms approach a specific value as the sequence progresses. In our text, we speculated on the existence of a limit for the sequence given by the recurrence relation \( a_{n+1} = \sqrt{1+a_n} \). A sequence that converges has great significance because the limit can sometimes reveal important intuitive or physical properties related to the problem at hand.

A convergent sequence is essential to various disciplines in mathematics, including analysis and number theory. It is particularly relevant when discussing infinite series where determining convergence is crucial to understanding the behavior of the series in question.

Golden Mean

Aside from the convergence topic, the exercise leads us to the golden mean, also known as the golden ratio. This intriguing number appears when we solve for the limit of our sequence and reveals itself to be \( (1+\sqrt{5})/2 \) approximately 1.61803398875. The golden mean has been studied for its unique properties and its occurrence in various aspects of nature, art, and architecture. It is particularly known for representing an 'ideal' aesthetic ratio.

The golden mean is not just a static number; it emerges in countless geometric constructions, algebraic processes, and even in the growth patterns of plants. Understanding this number can provide deeper insight into the natural world and the mathematics that describe it. Its relation to our sequence further exemplifies its pervasive presence in mathematical phenomena.

Limits of Sequences

To understand the concept of limits of sequences, as approached in the repeated square roots problem, we must discuss the formal definition. In mathematical terms, if the terms of a sequence become arbitrarily close to a number \( L \) as the sequence progresses, we say that the sequence approaches the limit \( L \) as \( n \) approaches infinity, denoted as \( \lim_{n\rightarrow\infty} a_n = L \). Assuming that \( L \) exists for our sequence, we can make profound statements about the behavior of the sequence for very large \( n \), or even its applications to real-world contexts. This helps mathematicians and scientists make predictions and understand asymptotic behaviors.

Real-world applications such as signal processing, control theory, and even economics rely on understanding the limits of sequences to model stable systems or predict long-term behaviors.

Solving Quadratic Equations

When we look for the exact limit of the sequence defined by the recurrence relation \( a_{n+1} = \sqrt{1+a_n} \) in our problem, we essentially solve a quadratic equation. A quadratic equation is a second degree polynomial equation of the form \( ax^2+bx+c=0 \). In our case, through the substitution \( a_n = L \) as \( n \) approaches infinity, we derived the quadratic \( L^2 - L - 1 = 0 \). To solve it, we used the quadratic formula which results in two solutions, out of which we accept the positive one, yielding the golden mean as the limit of our sequence, because we were looking for a value that made sense in the context of our sequence being positive.

Solving quadratic equations is a fundamental skill in algebra, which has practical applications across sciences and engineering. Learning to solve these can help students tackle a variety of problems and understand concepts such as the golden mean within mathematical phenomena.